I just got a golden Magikarp from GTS Negotiations. I want to know how to breed it to have another golden magikarp, and if I do decide to evolve this magikarp into a gyarados, will it still be shiny? Will I be able to breed it if it's a shiny gyarados into a golden magikarp?

The Golden Magikarp, also known as the more simple 'Shiny Magikarp' is like any other shiny Pokemon - meaning that the chance of a Shiny Magikarp's offspring being shiny is still the dumb old 1 in 8,000 chance.

However, unless you use all possible methods to increase your chances of getting a shiny (I'll list them when I find them. xD) the lowest chance of finding a shiny through an egg is 1 in 1,000. That's eight times the luck.

But, since a Pokemon is shiny doesn't mean that it's offspring will be shiny (or, will have an advantage of being shiny). Sorry. :(

Hope I helped. :)
Source: My wonderful self.

EDIT: To reach a 1 in 1,000 chance you will need to do the Masuda Method (breed two Pokemon that are from different countries, like an American Magikarp and a German Poliwhirl or something - this increases your chances to 3 in 4,096 (around 1 in 1,365 with a bunch of decimal places after)

The Shiny Charm which can be acquired by showing your completed Unova Pokedex to Cedric Juniper - he will give you the Shiny Charm which would increase your chances to 1 in 2,730. When you acquire this, you will always have this, meaning it will always be 1 in 2,730 and not 1 in 8,192.

Combining these two methods together creates the chance of a 1 in 1,024 chance to get a shiny egg. Eight times better the chance than normal.

Breeding a shiny Pokemon does not increase the chance of getting another shiny Pokemon. And yes, if the magikarp evolves, it will still be shiny. A way to breed for another shiny magikarp is by breeding that magikarp with a Pokemon from a different language aka masuda method or just regular breeding it a million times.

Breeding a shiny Pokemon does not increase that Pokemon's chance of being shiny. The hatched magikarp will have a 1/8192 chance of being shiny. If you evolve the magikarp, it will still be shiny, as evolution does not affect shininess.